In the present work we were motivated by recent interest in different forms of
helimagnetism [1]}[2]}, which can be explained in
terms of a chiral mechanism proposed by Dzyaloshinskii and Moriya (DM) a long
time ago [3]}. According to this DM mechanism, interactions are
restricted to first neighbors, but with vector spin variables on each lattice
site. Modulation effects come from the competition between the usual
Heisenberg exchange, of scalar nature, and an additional exchange associated
with the vector products of the DM interactions. Different formulations of
spin models including competing exchange and DM interactions have indeed been
used to account for the monoaxial helical patterns of magnetic compounds
[4]}[5]}. We were also motivated by early work
for special clock models with chiral interactions, which have been
investigated by different techniques [6]}[7]}[8]}. In contrast to the ANNNI model, elementary lattice model
systems associated with the DM mechanism have not been subjected to more
detailed statistical mechanics calculations. We then believe that there is
still room to analyze some simple and schematic version of these spin models
with DM interactions, and to show that they are capable of displaying a phase
diagram with many sequences of modulated structures.
